Please  refer to  Low Dimensional Topology blog.

virtually Haken conjecture   states that every compact, orientable, irreducible three-dimensional manifold with infinite fundamental group is virtually Haken.

Virtual Haken 猜想  设$$M$$ 为紧致的可定向的不可约的基本群无限的$$3$$-流形, 则$$M$$有一个$$Haken$$ 流形的有限覆盖.

$$3$$-流形的有限覆盖一直是$$3$$-流形拓扑理论中重要但进展缓慢的课题.

Ian Agol (UC Berkeley) announced a proof  of  the  Wise’s  Conjecture  on March 12, 2012  when he was speaking  in a  seminar lecture  at  the Institut  Henri Poincaré.  In  particular, this implies  the Virtually Haken Conjecture.  His proof is based on joint work with Daniel Groves (UI Chicago) and Jason Manning (SUNY Buffalo).  It makes heavy use of the work of Dani Wise (McGill) on the Virtually Fibred Conjecture, as well as the proof of the Surface Subgroup Conjecture by Jeremy Kahn (Brown) and Vlad Markovic (Caltech). The Proof was subsequently outlined in three lectures March 26 and 28th at the Workshop on Immersed Surfaces in $$3$$-Manifolds at the Institut Henri Poincaré. A preprint of the claimed proof  has been posted on the ArXiv , pdf .

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